Mathematics – Number Theory
Scientific paper
2009-08-29
Mathematics
Number Theory
7 pages, no figures. To appear, Bull. Aust. Math. Soc. Minor corrections
Scientific paper
Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < \P >: n \leq x} \frac{\mu(n)}{n}$ are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to $\prod_{p \in \P} (1 - \frac{1}{p})$ (the case when $\P$ is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of non-trivial zeroes and poles of the associated zeta function $\zeta_\P(s) := \prod_{p \in \P} (1-\frac{1}{p^s})^{-1}$ on the line $\{\Re(s)=1\}$. As equivalent forms of the first inequality, we have $|\sum_{n \leq x: (n,P)=1} \frac{\mu(n)}{n}| \leq 1$, $|\sum_{n|N: n \leq x} \frac{\mu(n)}{n}| \leq 1$, and $|\sum_{n \leq x} \frac{\mu(mn)}{n}| \leq 1$ for all $m,x,N,P \geq 1$.
No associations
LandOfFree
A remark on partial sums involving the Mobius function does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A remark on partial sums involving the Mobius function, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A remark on partial sums involving the Mobius function will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-597415